Problem 51 - College Physics

Question

\(\bullet\) Tuning a violin. A violinist is tuning her instrument to concert \(\mathrm{A}(440 \mathrm{Hz}) .\) She plays the note while listening to an electronically generated tone of exactly that frequency and hears a beat of frequency 3 \(\mathrm{Hz}\) , which increases to 4 \(\mathrm{Hz}\) when she tightens her violin string slightly.(a) What was the frequency of her violin when she heard the 3 -Hz beat? (b) To get her violin perfectly tuned to concert \(\mathrm{A}\) , should she tighten or loosen her string from what it was when she heard the 3 -Hz beat?

Step-by-Step Solution

Answer

(a) 437 Hz (b) Tighten the string.

Step 1: Understanding the Problem

The violinist is tuning her violin to match the standard A note, which is set at 440 Hz. She hears beats when her violin frequency differs from this standard frequency. We need to determine the initial frequency of her violin when she hears a 3 Hz beat and suggest whether she should tighten or loosen her string.

Step 2: Understanding Beats Frequency

When two frequencies are close but not identical, they produce a phenomenon called 'beats.' The beat frequency is the absolute difference between the two frequencies. In this case, the beat frequency is given as 3 Hz and then increases to 4 Hz when the string is tightened.

Step 3: Mathematical Concept of Beats

The beat frequency formula is \( f_\text{beat} = |f_1 - f_2| \), where \( f_1 \) is the frequency of the standard A (440 Hz), and \( f_2 \) is the frequency of the violin. Initially, the beat frequency is 3 Hz, which gives us \( |440 - f_2| = 3 \).

Step 4: Solving for the Initial Frequency

Solving the equation \( |440 - f_2| = 3 \) gives two possibilities: \( f_2 = 440 + 3 = 443 \) Hz or \( f_2 = 440 - 3 = 437 \) Hz. Thus, the initial frequency of the violin could either be 443 Hz or 437 Hz.

Step 5: Analyzing Beat Frequency Change

When the string is tightened slightly and the beat frequency increases to 4 Hz, it indicates that the frequencies are getting further apart. If \( f_2 = 443 \) Hz initially, tightening would increase the frequency further, causing a larger beat frequency (not matching the 4 Hz observation). Hence, it must be 437 Hz.

Step 6: Direction for Adjustment

Since the initial frequency was 437 Hz, which is less than 440 Hz, tightening the string will increase the frequency towards 440 Hz, reducing the beat frequency towards zero as required for perfect tuning.

Key Concepts

Beat Frequency

When a violin is out of tune, and you hear a pulsing or throbbing sound as you play, you're experiencing what's called "beat frequency." This phenomenon occurs when two sound waves with slightly different frequencies interfere with each other. Instead of producing a smooth, clear tone, the sound waves combine to produce a fluctuating volume effect. This is called a "beat," and its frequency is simply the difference between the two original frequencies.
For example, when the violinist played the note and heard a 3 Hz beat, it meant her violin's string was either vibrating at 437 Hz or 443 Hz if the reference was 440 Hz. The beat serves as an indicator of how far off a string is from the correct pitch. Fixing the issue involves changing the string's tension to eliminate this beat.
Typically, musicians listen closely to these beats during tuning and adjust their instrument until the beats gradually disappear.

Concert Pitch

Concert pitch refers to a standard reference frequency that musicians use for tuning their instruments. The most common standard used is the note A above middle C, set to 440 Hz. This means that in order for instruments like violins, pianos, and other members of an orchestra to be in harmony, their tuning needs to match this specific pitch.
Tuning to concert pitch ensures a unified sound in performances and recordings. It's especially important in orchestras, where many different types of instruments need to blend seamlessly together.

• The concert pitch of 440 Hz is often considered the "standard pitch" because it provides a common ground for tuning various instruments.
• By referencing this frequency, musicians can ensure that they are consistently in tune with western music traditions.

Frequency Adjustment

Frequency adjustment is crucial when tuning instruments to ensure they produce the correct pitch. When frequencies are slightly off, adjustments are made by changing the tension on the instrument's strings. Tightening the string increases its frequency, causing the pitch to rise, while loosening decreases the frequency.
In the violin tuning process, careful attention to how the beat frequency changes with adjustments can guide whether to tighten or loosen the string. For the violinist in the exercise, since increasing tension widened the beat frequency discrepancy (from 3 Hz to 4 Hz), it became clear that her initial frequency was below the target 440 Hz. Thus, tightening was necessary to reach closer to the desired pitch.
Understanding which adjustments to make requires both a keen ear and a familiarity with how physical changes to the instrument affect sound production.

Music Theory

Music theory is the study of the practices and possibilities of music. Within it, concepts like pitch, intervals, and tuning systems are deeply explored, providing musicians with frameworks for understanding harmony and melody.

• Pitch is the perceived frequency of a sound, which means higher pitches correspond to higher frequencies.
• When instruments are out of tune, dissonances can be created, which sound unpleasant or "off" to our ears.

In tuning, music theory helps us understand why adjustments result in certain changes of sound. The mathematical relationship between notes, like the octave (doubling in frequency), helps musicians predict how small tunings affect overall harmony.
For instance, a note an octave above A (440 Hz) would be 880 Hz. Understanding these relationships allows musicians to fine-tune their instruments to achieve the ideal sound.